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Wednesday, December 16, 2009

Most of my readers know that my philosophy is to challenge ALL of our students more than we do at present. The following problem should not be viewed therefore as a math contest problem for middle schoolers; rather a problem for all middle schoolers and on into high school

List all 5-digit palindromes which have zero as their middle digit and are divisible by 9.

Comments:(1) Should you include a definition or example of a palindrome as is normally done on assessments or have students "look it up!"

(2) Is it necessary to clarify that we are only considering positive integers when we refer to a 5-digit number?

(3) What is the content knowledge needed? Skills? Strategies? Logic? Reasoning? Do these questions develop the mind while reviewing the mathematics? In other words, are they worth the time?

(4) BTW, there are ten numbers in the list. Sorry to ruin the surprise!

(5) How would this question be worded if it were an SAT problem? Multiple-choice vs. grid-in?

6 comments:

1) I'd ask who could explain what it meant. "Look it up" is something I'd beg off on as something I "don't have time for".

2)I don't think so

3) Content knowledge: divisibility by 9 rule

4) Wait, really? I have 9. Are you letting the number begin with 0? That typically wouldn't count as a five-digit number.

5) Not sure about the SAT but I'd ask how many rather than asking them to make a list. I would assume most of them *would* make a list, but knowing *to* make the list is part of what I want them figuring out.

Mathmom,Thanks for your thoughts about instructional decision-making. Did you remember to include both 90009 and 99099?I designed this problem so that it would be inviting to many students. On the surface it's fairly straightforward but I like to add a little twist for the super-quick student who might jump too quickly. Also, considering how many careless errors I make, I am gun-shy about giving my answer!

So what if we lift the restriction on the middle digit? (major extension. Changes focus)

Or if we ask for palindromes under 100,000? (small but important extension. Forces dealing with "middle," a brief additional search, and whether a 1-digit number is a palindrome, and whether 0 is a 1-digit number (or a "number" at all).

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SAT Math Tips

ZERO IS A 'WEIRDO'! (W)hole(E)ven(I)nteger(R)Rational/Real(DO) Cannot Divide by O!BUT Zero is NOT Positive and NOT Negative!

POSITIVE INTEGERS start from 1

PRIMES start from 2 (not 1)

INTEGERS can be NEGative (and zero!) as well as positive

MEMORIZE the formula for the nth term of an arithmetic sequence: a(n) = a(1) + (n-1)d.Example: Consider the sequence of positive integers which leave a remainder of 3 when divided by 4. What is the 100th term?Step 1: List the first few terms 3,7,11,15,... to see the pattern and recognize it is an arithmetic sequence.Step 2: Identify the values which are givenFirst term or a(1) = 3Common difference or d = 4Number of terms or position of desired term or n = 100Step 3: Substitute into formula and solvea(100) = a(1) + (100-1)(4) = 3 + (99)(4) = 399

Of course there are other ways to find the 100th term such as 100 x 4 - 1 but the formula is so useful for so many types of questions it is worth learning!

Know the above by heart and you are way ahead of the game! These facts will absolutely be needed on your next SAT or standardized test!

About Me

Recently retired math educator and Supervisor of Mathematics; 30 years experience as an Advanced Placement Calculus (BC) teacher; Former Author of Math Teachers of New Jersey Annual HS Math Contest; Former K-5 Chair of New Jersey Math Content Standards and Curriculum Frameworks; Former member of Math Item Review Committee for New Jersey High School Proficiency Assessment; Experienced SAT Math Instructor and author of SAT materials; speaker at many regional and national math conferences